Correlation functions inXYmodels and step free energies in roughening models

Abstract

A duality relation derived by José, Kadanoff, Kirkpatrick, and Nelson and by Knops is exploited to calculate properties of $\mathrm{XY}$ and roughening models from known properties of the dual models. It is shown that the correlation length in $\mathrm{XY}$ models is exactly given by ${\xi }^{-1\mathrm{}}=\tilde{\beta }\tilde{f}$, where $\tilde{f}$ is the free energy per unit length of a step and $\tilde{\beta }$ is the inverse temperature in the corresponding roughening model. Similarly, the exponent $\eta$ below $T_c$ in an $\mathrm{XY}$ model is given by $\eta ={\lim }_{\left|\overrightarrow{\mathrm{r}}\right|\to \infty }\frac{\tilde{\beta }{\tilde{F}}_1\left(\overrightarrow{\mathrm{r}}\right)}{ln\left|\overrightarrow{\mathrm{r}}\right|}$, where ${\tilde{F}}_1\left(\overrightarrow{\mathrm{r}}\right)$ is the free energy associated with two screw dislocations of unit strength and opposite sign separated by the distance $\overrightarrow{\mathrm{r}}$ in the corresponding roughening model.